Feedbacks of Vegetation on Summertime Climate …cliveg.bu.edu/download/manuscripts/wlwang03.pdftemperature, particularly when time lags are longer than 2 months (W1). That is, vegetation

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Feedbacks of Vegetation onSummertime Climate Variabilityover the North AmericanGrasslands. Part II: A CoupledStochastic ModelWeile Wang* and Bruce T. AndersonDepartment of Geography and Environment, Boston University, Boston, Massachusetts

Dara EntekhabiRalph M. Parsons Laboratory, Department of Civil and Environmental Engineering,

Massachusetts Institute of Technology, Cambridge, Massachusetts

Dong Huang and Robert K. KaufmannDepartment of Geography and Environment, Boston University, Boston, Massachusetts

Christopher PotterEcosystem Science and Technology Branch, NASA Ames Research Center, Moffett

Field, California

Ranga B. MyneniDepartment of Geography and Environment, Boston University, Boston, Massachusetts

Received 27 January 2006; accepted 2 May 2006

* Corresponding author address: Weile Wang, Department of Geography and Environment,Boston University, 675 Commonwealth Ave., Boston, MA 02215.

E-mail address: [email protected]

Earth Interactions • Volume 10 (2006) • Paper No. 16 • Page 1

ABSTRACT: A coupled linear model is derived to describe interactionsbetween anomalous precipitation and vegetation over the North AmericanGrasslands. The model is based on biohydrological characteristics in the semi-arid environment and has components to describe the water-related vegetationvariability, the long-term balance of soil moisture, and the local soil–moisture–precipitation feedbacks. Analyses show that the model captures the observedvegetation dynamics and characteristics of precipitation variability during sum-mer over the region of interest. It demonstrates that vegetation has a preferredfrequency response to precipitation forcing and has intrinsic oscillatory vari-ability at time scales of about 8 months. When coupled to the atmosphericfields, such vegetation signals tend to enhance the magnitudes of precipitationvariability at interannual or longer time scales but damp them at time scalesshorter than 4 months; the oscillatory variability of precipitation at the growingseason time scale (i.e., the 8-month period) is also enhanced. Similar resonanceand oscillation characteristics are identified in the power spectra of observedprecipitation datasets. The model results are also verified using Monte Carloexperiments.

KEYWORDS: Land–atmosphere interactions; Vegetation feedbacks; Soilwater balance

1. IntroductionIn the first part of this study (Wang et al. 2006, hereafter W1), statistical

techniques are used to detect and analyze the influence of vegetation on climatevariability over the North American Grasslands. Results indicate significantGranger causal relationships (Granger 1969; Granger 1980) from lagged anoma-lies of vegetation activity to variations of summertime precipitation and surfacetemperature, particularly when time lags are longer than 2 months (W1). That is,vegetation anomalies earlier in the growing season provide unique information (ina statistically significant fashion) regarding climate variability later in summer.The nature of these causal relationships suggests that if mean vegetation anomaliesare high (positive) during the preceding months and/or the vegetation anomaliesshow a decreasing trend, the climatic conditions during the following summermonths (July–September) will tend to be drier and warmer (W1). As suggested bythe sign of these climate–vegetation interactions, vegetation and precipitationanomalies are also found to have distinct oscillatory variability at growing seasontime scales (W1; also see below).

A physical mechanism was proposed to explain the statistical results of W1. Inan arid/semiarid environment, where soil water storage is limited, initially en-hanced vegetation may deplete soil water faster than normal and thus generatenegative soil moisture anomalies later in the growing season. The drier soil in turnreduces fluxes of water and latent heat to the atmosphere and leads to lowerprecipitation and higher temperatures. Because drier soils also force vegetation todecrease, such interactions may generate oscillatory adjustments in the two fields.Part of this mechanism has been suggested by previous studies (e.g., Heck et al.1999; Heck et al. 2001).

Conceptual models of vegetation–atmosphere interactions (e.g., Brovkin et al.1998; Zeng et al. 2002; Wang 2004) and soil moisture–atmosphere interactions


(e.g., Rodriguez-Iturbe et al. 1991a; Rodriguez-Iturbe et al. 1991b) have beenproposed previously by numerous researchers. Through simplified force-feedbackschemes, these models are able to analytically capture many of the complicateddynamical characteristics of the climate system. For instance, analyses of thesemodels indicate that vegetation–atmosphere interactions may lead to multiplestable equilibria of the climate system at long time scales (Brovkin et al. 1998;Zeng et al. 2002; Wang 2004). For each of these long-term equilibria, on the otherhand, vegetation–atmosphere interactions also act to resist small (and transient)disturbances to the system and thus to maintain its stability (Brovkin et al. 1998;Alcock 2003). From the perspective of dynamics, this stability suggests that suchshort-term (e.g., intraseasonal) vegetation–atmosphere interactions may be simpli-fied using linear approximations (Glendinning 1994; also see the appendix). Fol-lowing this line of thinking, therefore, this paper aims to develop a simple linearmodel to capture the physical mechanism hypothesized by W1.

The rest of the paper is arranged as follows. Section 2 presents a developmentof the stochastic model used in our study. Section 3 identifies the model andretrieves the necessary system parameters from the observational datasets; section4 analyzes the dynamic characteristics of vegetation in the model and discusseshow they may regulate the variability of precipitation; and section 5 conductsMonte Carlo experiments to further verify the role vegetation plays in influencingprecipitation. Section 6 presents the concluding remarks.

2. Model developmentFor the proposed mechanism governing vegetation–climate interactions in semi-

arid grasslands regions, three important components need to be considered: 1) thevariability of vegetation (V) driven by available soil moisture (S) (Woodward1987; Nemani et al. 2003); 2) the balance of soil moisture maintained by precipi-tation (P) and evapotranspiration (ET) (Wever et al. 2002); and 3) the couplingbetween soil moisture and precipitation (Koster et al. 2004). When anomalies ofthese fields are considered, these relationships may be formulated in the followinglinear difference equations (see the appendix for detailed discussions):

V�t = �V�t−1 + �S�t−1, (1)

S�t = S�t−1 + P�t − ET�t, (2)

P�t = �S�t−1 + �t , (3)

where V�, S�, P�, and ET� represent anomalies of the corresponding variables; �t inEquation (3) represents the external precipitation variability, which is assumed tobe a random process (i.e., white noise); and �, �, and � are constant parameters,each of which corresponds to a particular physical process. Specifically, � (0 <� < 1) is the persistence rate of vegetation variability, � [normalized differencevegetation index (NDVI)/water] represents the short-term response of vegetationto soil moisture anomalies, and � (0 < � < 1) indicates the proportion of soilmoisture that contributes to local rainfall.

To formulate the evapotranspiration (ET) term in Equation (2), we assume it isa sum of two components: the evaporation component directly related to soil


moisture (Yamaguchi and Shinoda 2002) and the transpiration component asso-ciated with vegetation variability. Therefore, we represent ET as

ET�t = �1 − ��S�t−1 +1

�V�t, (4)

where � and � are also two constant parameters. In this way Equation (2) can berewritten as

S�t = �S�t−1 + P�t −1

�V�t. (2�)

Physically, the parameter � (0 < � < 1) is the persistence rate of soil moisture. Thecoefficient 1/� represents the rate of transpiration associated with unit vegetationanomalies. It is written such that � has the same units as � (NDVI/water). As willbe discussed later, � can be interpreted as the (potential) long-term response ofvegetation to precipitation forcing.

Equations (1), (2�), and (3) represent a stochastic model that describes interac-tions among anomalies of vegetation, soil moisture, and precipitation in an arid/semiarid environment. Prototypes of these equations can be found in the literature.For example, a similar form of Equation (1) has been used to describe water-related vegetation dynamics in Zeng et al. (Zeng and Neelin 2000; Zeng et al.1999; Zeng et al. 2002); Equation (2) is derived from the “bucket model” of soilwater balance (e.g., Budyko 1956; Manabe 1969); and Equation (3) is based on theformulation of soil moisture–precipitation interactions in Rodriguez-Iturbe et al.(Rodriguez-Iturbe et al. 1991a; Rodriguez-Iturbe et al. 1991b). For this study,these relationships have been simplified such that they can be represented as linearequations with constant parameters [i.e., a linear time invariant (LTI) system].Nevertheless, the physical meaning of the model and its parameters remains clearfrom the discussion above. Detailed derivations of the model from representativeequations found in the literature are given in the appendix. It should also be notedthat in order to keep the model simple and the analysis feasible, we have neglectedthe influence of temperature variations in this scenario; however, analyses indicatethat inclusion of temperature in the model does not qualitatively influence theresults discussed below.

To facilitate the comparison with the observed covariability of precipitation andvegetation (W1), the model can be represented in a form that explicitly involvesonly V and P:

V�t = �� + � − ��V�t−1 − ��V�t−2 + �P�t−1, (5)

P�t =�� − ��

�V�t−1 −

��

�· V�t−2 + �P�t−1 + �t . (6)

Thus, Equation (5) describes the variability of vegetation as driven by precipitation(referred to as the open-loop model hereafter), Equation (6) represents feedbacksof land surface processes to the atmosphere (referred to as the feedback function),and together, Equations (5)–(6) are referred to as the closed-loop model. This formof the model will be discussed in the following sections.


3. Datasets and model identification3.1. Datasets

To retrieve the parameters of the model, we use datasets of precipitation (Xieand Arkin 1997) and NDVI (Tucker 1979; Myneni et al. 1997; Myneni et al. 1998;Zhou et al. 2001), which is a measure of vegetation activity. Both datasets providemonthly measurements at global scales over the period of 1982–2000, and theNDVI dataset is aggregated to match the resolution of the precipitation dataset.They are then compiled to form a panel for the North American Grasslands, whichincludes about fifty-one 2° × 2° grid points that have the biome type of grasslands(Friedl et al. 2002) and lie within 25°–55°N and 90°–130°W (W1). For bothdatasets, monthly anomalies are calculated relative to their 1982–2000 climatolo-gies (i.e., long-term mean seasonal cycles), respectively. More details about thesedatasets can be found in W1.

Another dataset used in this study is the Climate Prediction Center (CPC) U.S.UNIFIED Precipitation dataset (Higgins et al. 2000). This dataset providesmonthly (and daily) precipitation measurements from 1948 to 1998 for the con-tinental United States at a spatial resolution of 0.25° × 0.25°. There are about 2700grid points in the domain of the North American Grasslands. Monthly anomaliesof this dataset are calculated relative to their 1948–98 climatologies. These anoma-lies are used to calculate the mean power spectra of precipitation variations overthis region, which will be discussed later in the paper.

3.2. Model identification

The form of Equations (5) and (6) readily allows their parameters to be esti-mated from the observational datasets by the methods of regression. For thispurpose, we rewrite the equations as

V�t = A1V�t−1 + A2V�t−2 + A3P�t−1 + �1t , (5�)

P�t = B1V�t−1 + B2V�t−2 + B3P�t−1 + �2t , (6�)

where As and Bs represent the corresponding regression coefficients.We utilize two algorithms, namely, the ordinary least squares (OLS) method and

the out-of-sample forecast method (see appendix A of W1), to estimate the re-gression coefficients of Equations (5�)–(6�) and to test their statistical significancelevels. Different from the OLS method, the method of out-of-sample forecast isparticularly devised for panel data (Granger and Huang 1997): it examines howwell the statistical model can predict the climate/vegetation variability at gridpoints that are not included in the regression (W1); also, the testing statistics areconstructed based on the accuracy of the predictions rather than the r2 of theregressions (W1; Diebold and Mariano 1995). As such, this method helps to avoidspurious fits of the statistical model to the data (Granger and Huang 1997). Nev-ertheless, consistent regression results are obtained via the two algorithms. For thesake of simplicity, therefore, we mainly discuss the results obtained using the OLSmethod.

Table 1 gives the regression results of Equations (5�) and (6�). Generally, resultsindicate that the open-loop model captures the variability of NDVI anomalies


(Table 1a). The r2 statistics of the regressions of Equation (5�) are above 60% forall the months from June to October, and as high as 80% in August. All estimatesof the regression coefficients (As) are significant (p < 0.05) and physically con-sistent during this period. For example, the signs associated with these coefficientsshow a regular pattern through the season, such that A2 is always negative whileA1 and A3 are always positive. Because A2 represents –�� [Equation (5)], negativevalues of this coefficient are consistent with the derivation of the model.

Compared with the open-loop model, identification of the parameters of thefeedback equation [Equation (6�)] is much more difficult. The regression results(Table 1b) generally indicate low r2 values (2%–9%), and the estimates of thecoefficients (Bs) are also scattered: August is the only month in which the threecoefficients are statistically significant and have the expected sign (Table 1b).These results reflect the fact that land surface feedbacks are generally weak andcan be masked by the stronger internal variability of the atmosphere. Nevertheless,the results of Table 1b still provide a way to estimate all the system parameters.

Ideally, the five system parameters can be retrieved from the coefficients inEquations (5�) and (6�) using the following relationships:

� = A3, (7a)

� = B3 = �B2�A2, (7b)

� = A1 − ��B1, (7c)

� = −A2��, and (7d)

� = �� + � − A1�. (7e)

Here, the parameter � can only be estimated from the feedback equation [Equation(6�)]. Based on Equation (7b) and the results of Table 1b, this parameter is

Table 1. Regression coefficients for (a) Equation (7) and (b) Equation (8) estimatedfrom observed precipitation and vegetation (NDVI) for the North American Grass-lands. Only regression coefficients that are statistically significant at 95% level areshown here.

(a)

Month A1 A2 A3 r2

Jun 1.059 −0.593 0.009 0.61Jul 1.008 −0.533 0.009 0.64Aug 0.943 −0.424 0.014 0.80Sep 0.762 −0.193 0.014 0.77Oct 0.980 −0.398 0.007 0.70

(b)

Month B1 B2 B3 r2

Jun 0.326 0.09Jul 3.236 0.02Aug 2.144 −4.306 0.125 0.06Sep −2.225 0.1417 0.03Oct −8.725 0.07


estimated to be about 0.13–0.15 for August and September, and may be as high as0.33 in June.

The other four parameters (�, �, �, and �) are contained in both the open-loopmodel and the feedback equation. Because the influence of precipitation on veg-etation variability is much stronger than the influence of vegetation upon precipi-tation, we try to estimate these parameters mainly based on the regression resultsof the open-loop model (Table 1a). However, because Equation (5�) has only threecoefficients (A1–A3), the four parameters are underdetermined by the open-loopmodel itself. To fully retrieve them, therefore, we either need to take into accountthe feedback equation [Equation (6�)] or make an a priori estimate for one of theparameters (see below).

Table 2 shows the estimated values of the five system parameters based on theregression results for August, in which the precipitation–vegetation interactionsare most evident (Table 1). As shown, the estimated persistence rate of NDVIanomalies (�) is about 0.70, and the persistence rate of soil moisture (�) is about0.60 (Table 2). We also make estimates for these parameters by arbitrarily setting� to 0.6∼1.0 at an interval of 0.1; correspondingly, � decreases from about 0.7 toabout 0.4. Nevertheless, we find that these different estimates do not affect theoverall characteristics of the model because the combination of the two will stillgive the same coefficients for Equation (5) and similar coefficients for Equation(6). Therefore, we use the estimates in Table 2 as typical parameters to illustratethe characteristics of the model in the following sections. When it is necessary, wewill also discuss how different values of the parameters may induce additionalvariability as a means of interpreting the characteristics in a physically meaningfulway.

4. Model analysisThe regression results of the last section (Table 1) indicate that the open-loop

model captures the variability of vegetation driven by precipitation. First, we willdiagnosis this model in order to investigate and interpret the dynamics of thevegetation variability. Then we will study how vegetation signals may modulatethe variability of precipitation when they propagate to the atmosphere by analyzingthe feedback equation [Equation (6)]. Finally, we will try to identify some of thesimulated characteristics within the corresponding observed precipitation datasetsin order to provide additional evidence for the hypothesized vegetation–precipitation feedbacks contained in the model.

4.1. Open-loop model

As a first check for the open-loop model, we use it to simulate the observedvegetation variations. That is, given known observed precipitation information and

Table 2. System parameters used in the open- and closed-loop models for theNorth American Grasslands.

� � � � �

0.703 0.014 0.603 0.039 0.125


two initial values of NDVI, Equation (5) is used to predict observed vegetationvariations for the rest of the growing season. Figure 1 shows the results of thesimple simulation. Generally, the simulated NDVI evolves with the observed timeseries, resulting in good agreement between the two (Figure 1). As shown, al-though the input precipitation signal is fairly noisy, vegetation over the midlatitudegrasslands has lower-frequency variations and an apparent oscillatory component,as suggested by W1 (Figure 1).

These dynamic characteristics of the model can be quantified by the frequencyresponse functions of the system (Figure 2), which describes the magnitudes andphase angles of the outputs (NDVI) relative to the sinusoidal inputs (precipitation)at different frequencies. For the convenience of comparison, the gain function(Figure 2a) is normalized by the input gain factor [i.e., � in Equation (5)], such thata 0-dB value suggests that unit variance in precipitation generates unit variance inNDVI. (Such normalization will be used for discussions throughout the paperunless otherwise specified.)

The normalized gain function (Figure 2a) indicates “red” responses of NDVI toprecipitation forcing. The magnitude of NDVI is positive (∼6 dB) at longer timescales (i.e., periods longer than 8 months) but decreases to negative values (about

Figure 1. (top) NDVI anomalies simulated with the open-loop model (upper, dark),and the observed NDVI (upper, gray). (bottom) Precipitation anomaliesused in the simulation. The simulation uses the first two observed NDVIvalues of each growing season as initial values and is forced by theprecipitation anomalies during the course of the season.


−7 dB at the 2-month period) as the time scales become shorter. Correspondingly,the phase lag between NDVI and precipitation is small at longer time scales butdecreases to −180° at the 2-month period (the negative sign indicates that NDVIlags precipitation). The transitions of the magnitude and the phase functions occurat about the 8-month period, where the magnitude peaks (∼8 dB) and the phase lagis about −90° (Figure 2), indicating that the system has intrinsic oscillatory vari-ability at this frequency range (Reid 1983). Figure 2 also indicates that the spectralcharacteristics of the model (line plots) are consistent with those independentlyestimated from observations only (dot plots; also see W1).

Explanations for these frequency characteristics are provided by the derivationof the model. For example, Equation (1) indicates that vegetation has a “memory”(given by the persistence rate �) of soil moisture variations, and soil moisture itselfrepresents an accumulating process of precipitation [Equation (2)]. Both mecha-nisms act as low-pass filters to remove transient variations from the input precipi-tation signals. In addition, Equation (1) indicates that soil moisture surplus pro-motes vegetation growth; however, Equation (2) indicates that vegetation depletessoil water. The interaction between these two equations therefore can lead tooscillatory adjustments in the vegetation field.

Figure 2. System response functions of NDVI as driven by precipitation (open-loopmodel). (top) Gain functions; (bottom) phase function. The discrete points(“o”) show the estimates from the observed data (taken from W1).


A quantitative interpretation of the red responses of vegetation and its oscilla-tory variability can also be obtained by analyzing the analytical solution of themodel explicitly. As represented by Equation (5), the open-loop model is essen-tially a second-order difference equation. General solutions to such an equation aredetermined by its two characteristic roots (or eigenvalues):

�1,2 =�� + � − �� ± ��

2, (8)

where

� = �� + � − ��2 − 4��. (9)

When the characteristic roots have complex components (� < 0), this system willhave oscillatory variability.

With the estimated �, �, �, and � values (Table 2), it can be shown thatEquation (8) gives two complex characteristic roots at exp(−0.43 ± i0.76). Theimaginary part of the exponential indicates an oscillation frequency at 0.76 (radian/month), or equivalently at a period of 8.25 months, which is consistent with theestimated value here and in W1. The real part of the exponential is negative andindicates that the system is stable. Hence, the magnitude of the oscillatory varia-tions will decay at the rate of e−0.43 (i.e., e−1/2.3, or 0.65) per month; in other words,the time constant of the decay process is about 2.3 months.

The oscillatory decaying adjustments of the open-loop model are represented byits impulse-response function (Figure 3), which examines the output signals of thesystem when a unit precipitation anomaly is input at the initial time (i.e., P � 1at t � 0, and P � 0 for t > 0). Figure 3 shows that the positive response of

Figure 3. The oscillatory decaying process of NDVI’s response to an impulse input ofprecipitation at time t = 0 (open-loop model). The magnitude of NDVI isnormalized by the factor �.


vegetation arises in the next month (t � 1) and lasts for 4 months (with decreasingmagnitudes); the response then switches to the negative phase (t � 5 ∼ 8),representing the first cycle of the oscillation (Figure 3).

The above analyses all suggest that the model system has a preferred responseat low frequencies and that vegetation–soil moisture interactions within the modelpromote oscillatory behavior. These characteristics of vegetation variability can inturn be physically interpreted using the system parameters (i.e., �, �, �, and �) ofthe model itself. To interpret the red spectra of vegetation, we first check thestability of the system (which as discussed above indicates a stable evolution asopposed to an unstable evolution). By plugging two constant values for precipi-tation (Ps) and vegetation (Vs) into Equation (5), we find that

V�s = �*P�s, (10a)where

�* =�

�1 + �1 − ��1 − ��. (10b)

Here, �* describes the long-term relationship between vegetation and precipita-tion, and it is related to the system parameter �. To see this, note that �* is equalto � when � is 1 [Equation (10b)]. Because a value of 1 for � means that all waterloss is transpired through vegetation [Equation (2�)], � describes the potentiallong-term response of vegetation to precipitation, that is, a situation in whichvegetation continues to grow until it utilizes all available moisture. On the otherhand, as indicated by Equation (1), the rate of vegetation production is determinedby the short-term response coefficient �. Therefore, the ratio of �/� gives ameasure of the rate at which vegetation reaches its (potential) equilibrium withprecipitation. Given the estimates of the system parameters (Table 2), the value of� (0.014) is about one-third of � (0.039) and less than one-half of �* (∼0.030).This means that it takes some time for vegetation to fully respond to precipitationforcing. Therefore, vegetation responds more readily to persistent precipitationvariability at longer time scales, producing the red system response function.

In addition we can use Equation (9) to investigate the oscillatory variability ofthe open-loop system, which is determined by the values of �, �/�, and �. Asdiscussed above, when �/� is high, vegetation production is fast and soil water isconsumed rapidly. At the same time, because persistent vegetation anomalies alsoconsume water, vegetation must decay quickly (i.e., � must be small) to avoidnegative soil moisture anomalies from being generated. Clearly this process is alsoregulated by the soil moisture persistence rate �, which indicates the proportion ofsoil moisture subject to the influence of vegetation (see below).

We use � to normalize � and �/�, and rewrite Equation (9) as follows:

� = �2��1 + �* − �*��*�2 − 4�*�, (9�)where

�* =�

�, �*��* =

��

�.

As such, the oscillatory/nonoscillatory domain of the system can be determined onthe �* − �*/�* plane (Figure 4). To simplify the discussion, we first consider the


case when � is 1, and the values of �* and �*/�* (the same as � and �/�) arebetween 0 and 1. In these ranges, the system is nonoscillatory only if vegetationhas short memory (e.g., �* is close to 0) or it reaches its steady state slowly enough(i.e., �*/�* is close to 0; Figure 4). When � is less than 1, the impact of vegetationon soil moisture variability decreases. Therefore, the nonoscillatory domain is alsopresent in the range where �* or �*/�* is larger than 1 (Figure 4). In either case,however, Figure 4 indicates that for most values of the system parameters, oscil-lations are likely to be an intrinsic property of the soil moisture–vegetation system.

4.2. External disturbance to vegetation

In addition to precipitation, other external disturbances can also influence veg-etation variability. Such disturbances may include climate factors (e.g., tempera-ture, solar radiation, etc.) or nonclimatic factors (e.g., fire, pest, and human ac-tivity). To account for the influence of these processes, we introduce an externalinput term et to Equation (1), such that

V�t = �V�t−1 + �S�t−1 + et. (1�)

Figure 4. Subdomains of oscillatory (gray) and nonoscillatory (white) variability ofthe open-loop model on the �*−�*/�* plane [Equation (9�)]. The smallcircle (“o”) indicates the location of the observed system on this plane.


If we assume that et is independent of the precipitation forcing, the variability ofvegetation induced by et can be described by

V�t = �� + � − ��V�t−1 − ��V�t−2 + et − �et−1. (5�)

Because Equation (5�) has the same characteristic equation as Equation (5), thegeneral solutions to this equation have the same properties (i.e., stability andintrinsic oscillations) as discussed before. However, because there is no input ofwater in Equation (5�), the negative effects of et on soil moisture will generateopposing vegetation anomalies [i.e., �et−1 in Equation (5�)] in the followingmonths to counter any initial disturbances. This is illustrated by the impulse-response function (Figure 5). As shown, when et induces a unit positive vegetationanomaly at the initial time (t � 0), it rapidly decreases to a negative value in 2months (t � 2) and remains negative in the following four months (t � 2 ∼ 5).

Equation (5�) allows us to explicitly investigate the vegetation variability that isresponding to soil moisture variations as opposed to precipitation variations. Todenote the fraction of the vegetation variations that is related to soil moisture, wedefine an additional variable, V�soil, such that

V�soil = V�t − et. (11)

The spectral responses of V�t and V�soil to external disturbances of et are illustratedby the corresponding system functions (Figure 6). As shown, the magnitude of V�tis slightly negative at low frequencies (e.g., −1.4 dB at the 64-month period); at thesame time, the magnitude of V�soil is slightly positive (about 2 dB at the 64-monthperiod), but the phase angle is 180°, just opposite that of V�t . These results suggestthat if vegetation growth is initially positive (but there is no corresponding increasein the water supply), the soil will become drier and generate opposing vegetation

Figure 5. The oscillatory decaying process of NDVI’s response to an impulse input ofexternal vegetation disturbance at time t = 0 (open-loop model).


anomalies (V�soil) to cancel the input. As a result, we get an output of V�t lower(−1.4 dB, or 0.85) than the input.

4.3. Closed-loop modelBecause interactions between vegetation and soil moisture have distinct dy-

namic characteristics (e.g., Figure 2 and Figure 6), when such signals feed back tothe atmosphere [Equation (3) or Equation (6)], they will leave their spectral sig-natures on the variability of precipitation (Delire et al. 2004). This hypothesis isverified by the spectral characteristics of the closed-loop model.

Figure 7 shows the frequency-response functions of the output precipitation[i.e., P�t in Equation (6)] relative to the “original” precipitation forcing [i.e., �t inEquation (6)]. Overall, it indicates slight but clear red variability of P�t . Themagnitude is slightly enhanced (0.7 dB) at low frequencies but damped (−0.6 dB)at high frequencies, and the transition occurs at about the 8-month period, wherethe magnitude has a peak of about 1.5 dB (Figure 7). The phase function indicatesthat P�t varies almost in phase with �t at both high and low frequencies, and thephase lag is also small at transition frequencies (∼13°; Figure 7).

Figure 6. System response functions of NDVI as driven by external vegetation dis-turbances (open-loop model). The solid line shows the total output of NDVI(Vtotal), while the dashed line shows the portion of NDVI variations asso-ciated with soil moisture (Vsoil).


The frequency responses of P�t in Figure 7 have a clear association with thecorresponding characteristics of vegetation (Figure 2). Qualitatively, as vegetationvaries synchronously with precipitation at low frequencies (Figure 2), the corre-sponding transpiration anomalies induce water vapor anomalies with the “correct”sign to reinforce the variations of precipitation (Figure 7); on the other hand,because vegetation lags precipitation by almost 180° at high frequencies (Figure2), the corresponding transpiration anomalies induce water vapor anomalies withthe opposite sign, which in turn cancel a portion of the precipitation variability(Figure 7). At the intermediate frequencies where the land surface processes haveintrinsic oscillations, the oscillatory variability of precipitation is further enhanced(Figure 7).

Frequency-response functions are also calculated for precipitation as it is drivenby external vegetation disturbances [i.e., et in Equation (1�) or Equation (5�);Figure 8]. Again, the gain function of precipitation shows a red spectrum that hasenhanced (10 dB) magnitudes at low frequencies and damped (about −10 dB)magnitudes at high frequencies, with the transition occurring at about the 8-monthperiod (Figure 8). The phase function indicates that the responses of precipitationare opposite to the external forcing et at low frequencies, which is related to the

Figure 7. System response functions of precipitation as driven by external precipi-tation variations (closed-loop model). (top) Normalized gain functions;(bottom) phase function.


corresponding characteristics of the open-loop model (Figure 6). For instance,positive vegetation anomalies (without corresponding water input) will tend todecrease soil moisture (i.e., “Vsoil” of Figure 6) and as such they will have negativeimpacts on precipitation as well (Figure 8).

Together, the results of Figures 7 and 8 indicate that vegetation feedbacks addadditional variability to precipitation such that it has red spectra as well. To verifythis feature in the observations, we calculate power spectra of observed precipi-tation anomalies based on the CPC U.S. UNIFIED Precipitation datasets (Higginset al. 1996; Higgins et al. 2000). Before the power spectra are calculated, precipi-tation anomalies at each grid point are normalized to have unit variance, such thatthe spectra of all the grid points can be averaged to represent the mean powerspectra for the North American Grasslands (Figure 9, dark solid line). As shown,the mean power spectra of the observed anomalies have higher magnitudes (about−4 dB) at time scales longer than 16 months, but lower magnitudes (about −5.5dB) at periods of 4 months or shorter (Figure 9, dark solid line). The standarddeviations of the mean power spectra (averaged over 2700 grid points) are about0.03 dB at all frequencies; therefore, a magnitude difference of 1.5 dB indicatesred spectra of precipitation variations in a statistically significant fashion.

The observed red spectra of precipitation anomalies can be reproduced by model

Figure 8. System response functions of precipitation as driven by external distur-bances of vegetation (closed-loop model).


simulations (Figure 9, dash lines). Results indicate that when the model uses afeedback coefficient of �/2 (Table 2), the power spectra of the simulated precipi-tation (Figure 9, dark dash line) show a similar magnitude range as the observedvalues (Figure 9, dark solid line), while the spectra of the random inputs areessentially “flat” (Figure 9, gray dash line). When the model uses the full value of�, the output precipitation has larger magnitude differences between high and lowfrequencies (not shown). The overestimated red spectra of the model simulationsmay be due to the fact that the strength of land–atmosphere interactions in theobserved system has monthly/seasonal variability (W1), which is not included inthe model (here all values are constant through the entire simulation). For the samereason, the model may also overestimate the spectral magnitudes at 8–16-monthtime scales (Figure 9).

The red power spectra of the observed precipitation shown in Figure 9 do notrule out the possibility that they may be induced by other external processes atinterannual or longer time scales (e.g., ocean–atmosphere interactions). To addressthis problem, we divide the observed monthly precipitation anomalies into grow-ing season months (GS; March–October) and nongrowing season months (NonGS;September–April), and calculate the power spectra for the two time windowsseparately (Figure 10, top). The two 8-month periods are defined to facilitate thecalculation, and the overlap between them does not substantially influence theresults (not shown). For comparison, we also divide the model simulations and therandom inputs into 8-month segments and recalculate their power spectra, respec-tively (Figure 10, bottom).

As shown (Figure 10, top), the power spectra of the observed GS precipitationanomalies have higher magnitude (−4.7 dB) at the 8-month period and have lowermagnitudes (about −5.2 dB) at the 4-month or shorter time scales, which is con-sistent with the red characteristics of the full spectra in the same frequency ranges(Figure 9; also shown in gray solid line in Figure 10, top). On the other hand, the

Figure 9. Mean power spectra of observed precipitation and model simulations.“Obs”: observations compiled from the CPC U.S. UNIFIED precipitationdatasets (Higgins et al. 2000); “Simu”: simulations with the closed-loopmodel with the feedback coefficient set to �/2; “Ctrl”: random inputs usedfor the model simulation.


power spectra of observed NonGS precipitation anomalies have almost the samemagnitudes (about −5.1 dB) at periods of 8, 4, and 8/3 months, while there is amagnitude drop (about −5.5 dB) at the 2-month period (Figure 10, top). Comparingthe spectral characteristics of the observed precipitation anomalies with those ofthe model simulations, it indicates qualitative agreement between the observed GSanomalies and the simulated precipitation, as well as qualitative agreement be-tween the NonGS anomalies and the random inputs (Figure 10, bottom). There-fore, these results suggest that land surface feedbacks during the growing seasonrepresent an important source for the red spectra of the precipitation anomaliesshown in Figure 9, which is not present during the nongrowing season.

However, results of Figure 10 cannot be uniquely attributed to the influence ofvegetation upon precipitation variability. In fact, the feedback function of Equation

Figure 10. (top) Power spectra of observed precipitation (taken from the CPC UNI-FIED datasets) during the GS (March–October, dark solid line) and theNonGS (September–April, dark dashed line). The gray line shows thecontinuous power spectra for the observations as in Figure 9. (bottom)Power spectra of simulated precipitation (solid lines) and the randominputs. The dark lines show values calculated based on 8-month seg-ments of simulations and controls, and the gray lines are the correspond-ing continuous spectra from the simulations as shown in Figure 9.


(6) is not the only process that can lead to the red shift in the power spectra ofprecipitation. For example, removing the vegetation terms from Equation (6) gives

P�t = �P�t−1 + �t , (12)which can also induce red spectral characteristics of precipitation that are similarto those observed (not shown). Equation (12) can be understood as a direct inter-action between soil and the atmosphere (e.g., via evaporation), without any influ-ence of vegetation upon soil–atmosphere water exchanges. Theoretically, the prin-cipal difference between Equation (6) and Equation (12) is that vegetation feed-backs [Equation (6)] can introduce oscillatory variability in precipitation (e.g.,Figure 7 and Figure 8), while direct soil moisture–precipitation variations [ascontained in Equation (12)] cannot. However, because vegetation feedbacks arestrong only in certain seasons during the year (e.g., summer), identifying suchoscillatory characteristics directly from the power spectra of Figure 9 remainsdifficult. Therefore, other metrics are required to further analyze the role of veg-etation in influencing precipitation, which we discuss in the following section.

5. Model experimentsWe conduct Monte Carlo experiments to determine which candidate models

best capture the observed summertime precipitation–vegetation interactions. Thethree candidates are the open-loop model [Equation (5); “OPEN”], the closed-loopmodel with the feedback function described by Equation (6) (“VEG”), and theclosed-loop model in which the feedback function is given by Equation (12)(“AR1”). These models have the same model component to describe vegetationvariability [i.e., Equation (5)], but they differ from one another only in the waysthey treat precipitation: in the OPEN model precipitation variations are simplyassumed to be white noise; in the VEG model precipitation variability depends onthe preceding status of both precipitation and vegetation; and in the AR1 modelprecipitation is only related to its own preceding status, and not to vegetation.

In each model experiment, the external climate forcing and the external distur-bances to vegetation are represented by two random time series, which have aGaussian distribution and standard deviations of 0.7 (mm day−1, precipitation) and0.02 (NDVI), respectively. The simulated vegetation (V) and precipitation (P)are truncated (with the first 500 values discarded) to 11 628 samples (i.e., 12months × 19 yr × 51 pixels), which is the length of the panel data for the NorthAmerican Grasslands (W1). As such, these model simulations can be analyzed inthe same way that the observational data are analyzed (W1). The model experi-ments are repeated 10 000 times to estimate the mean state of each model as wellas the corresponding intraseasonal evolution.

We compare the model simulations with the observations using the autocorre-lations of monthly vegetation and precipitation anomalies, as well as the laggedcross correlations between them (Figure 11; correlations of the observed data areaveraged over July–September). Overall, all the models capture the observedvegetation variability. Simulations and observations show that the autocorrelationsof vegetation (NDVI) decrease from positive values (e.g., 0.7) to negative values(e.g., −0.2) as the time lag increases to 4 months (Figure 11a), which is an indicatorof oscillatory variability in the vegetation anomalies (W1). Also, all models and


observations indicate that the positive correlations between vegetation and thepreceding precipitation variations decrease as time lag increases (Figure 11b).

On the other hand, the models differ in describing the variability of precipita-tion. As shown in Figure 11c, autocorrelations of observed precipitation decreasefrom positive values (∼0.12, at the 1-month lag) to slightly negative values (−0.05,at lags longer than 3 months; Figure 11c, “OBS”). At the same time, cross cor-

Figure 11. Autocorrelations and lagged correlations of/between anomalies of veg-etation and precipitation. (a) Correlations between V(t ) and V(t–lag), (b)correlations between V(t ) and P(t–lag), (c) correlations between P(t )and P(t–lag), and (d) correlations between P(t ) and V(t–lag). VEG standsfor the closed-loop model with vegetation feedbacks [Equation (6)], AR1denotes the closed-loop model with precipitation related only to its ownpreceding status (but not vegetation), OPEN is the open-loop model,and OBS indicates observations that are averaged over the late summer(July–September).


relations between current precipitation (P�t ) and lagged vegetation (V�t−lag) havenegative values as time lag increases (Figure 11d, OBS). Of the model simulations,only the VEG model captures both these observed characteristics (Figure 11c andFigure 11d, VEG). In contrast, because precipitation in the OPEN model is ran-dom, it cannot be predicted either by preceding values of itself or those of veg-etation. As such, it has essentially zero autocorrelations and lagged cross corre-lations (Figure 11c and Figure 11d, OPEN). The AR1 model captures the decreas-ing trajectory of the autocorrelations of precipitation anomalies; however, theseautocorrelations converge to zero at the 3-month lag and do not become negative(Figure 11c, AR1). Also, vegetation in the AR1 model shows no information aboutfuture precipitation (Figure 11d, AR1). Together, these results indicate that theVEG model, which incorporates the feedback of vegetation upon the atmosphere,best captures the observed statistical precipitation–vegetation interaction charac-teristics.

Figure 11. (Continued )


It is noted that the correlation structures of model simulations are likely todeviate from the observations at long time lags (Figure 11). For example, simu-lations show that correlations between vegetation (V�t ) and lagged precipitation(P�t−lag) become slightly negative at the 5-month lag, while the observations do not(Figure 11b). Also, the negative correlations between simulated precipitation (P�t )and vegetation (V�t−lag) show a recovering tendency at lags longer than 3 months,while the observations indicate a further decrease (Figure 11d). These differencesmay reflect the seasonal variations of the precipitation–vegetation interactions.Additional analyses indicate that at the beginning of the growing season (e.g.,April and May) vegetation is less dependent on precipitation and is more stronglyrelated to temperature (not shown); in this sense, higher vegetation anomalies maybe associated with warmer temperatures but less rainfall during the cooler earlygrowing season months. Because both factors have negative impacts on soil mois-ture (e.g., Figure 3 and Figure 5), they suggest that higher vegetation in spring maybe followed by precipitation anomalies that are lower than expected in summer(negative correlation; Figure 11d, OBS). In the same way, higher precipitation inspring may precede higher-than-expected vegetation in summer (positive correla-tion; Figure 11b, OBS) because the consumption of soil water is delayed until laterin the growing season.

Finally, we verify these model simulations by testing Granger causal relation-ships from vegetation to precipitation, which examines whether lagged vegetationanomalies contain information to predict current precipitation variability (W1). Inparticular, for the case of 4-month lags, we test how the mean (V�mean) and the trend(V�diff) of vegetation anomalies over the preceding months contribute to suchrelationships (W1). From W1, V�mean and V�diff are defined by

V�mean = �l = 1

4

V�t−l�4, V�diff = �l = 1

2

V�t−l�2 − �l = 3

4

V�t−l�2. (13)

Of the total 10 000 experiments, the results indicate that 96% of VEG simulationshave a significant causal relationship from vegetation to precipitation, while thecorresponding numbers for both OPEN and AR1 models are about 4%. In addition,in 99% of the VEG experiments the sign associated with V�mean is negative, and in84% of the VEG experiments, the sign associated with V�diff is positive. Theseresults indicate that higher mean values or/and decreasing trends of vegetationvariations are likely to precede lower precipitation anomalies and hence are con-sistent with the findings of W1 based upon the observed data. Furthermore, it isfound that when the model (VEG) is simulated only with the external climateforcing, in which all the information of vegetation is derived only from precipi-tation, the significant causal relationships disappear (not shown). These experi-ments further indicate that a system in which there is negligible influence of vegetationon precipitation could not spuriously produce the observed results found in W1.

6. ConclusionsThis paper develops a simple stochastic model to quantitatively investigate

interactions between anomalous precipitation and vegetation over the NorthAmerican Grasslands. Based on characteristics of biohydrological processes in thissemiarid environment, the model has components to describe the water-related


vegetation variability, the long-term balance of soil moisture, and the local soilmoisture–precipitation feedbacks. These dynamic relationships are quantified byfive physically meaningful parameters, which describe the persistence rate ofvegetation, precipitation, and soil moisture, as well as the short-term/long-termresponses of vegetation to precipitation forcing. To facilitate the comparison withthe observational analysis (W1), the model is represented in two linear differenceequations, with a focus on how precipitation drives variations in vegetation (theopen-loop model), and how vegetation signals feed back to the atmosphere (thefeedback equation). This representation allows the system parameters to be readilyestimated from the observational data.

Analyses indicate that the open-loop model, which represents the variability ofvegetation driven by precipitation, captures two principal characteristics of theobserved vegetation dynamics. First, the model indicates that vegetation has a“red” response function to precipitation forcing; that is, vegetation responds moreactively to low-frequency precipitation variations but does not respond to rapiddisturbances. Second, the model indicates that the system has intrinsic oscillatoryvariability, which is induced by interactions between vegetation and soil moisture.With the estimated system parameters, this oscillation frequency is calculated to be0.76 rad month−1, representing a period of about 8 months, which is consistentwith the analysis of W1.

Analyses of the closed-loop model indicate that vegetation signals can impartsimilar spectral signatures on the variability of precipitation when they feed backto the atmosphere. The frequency-response functions of the model show that themagnitude of precipitation is slightly enhanced at low frequencies (e.g., periods8 months) but damped at high frequencies (e.g., periods 4 months), regardlessof the external forcing coming from the climate side or the vegetation side. Suchred frequency characteristics are verified by the power spectra of observed pre-cipitation datasets and the model simulations. The role of vegetation in influencingprecipitation variability is further verified by Monte Carlo experiments. Only whenthe vegetation feedback is incorporated can the model simulations reproduce char-acteristics of observed covariability between summertime precipitation and veg-etation anomalies, as represented by the lagged correlations and the Granger causalrelationships. These relationships indicate that higher mean values and/or decreas-ing trends of vegetation anomalies are likely followed by lower precipitationanomalies, in agreement with observed results (W1).

More important than just replicating the characteristics of the observed precipi-tation–vegetation interactions, the dynamic aspects of the model provide insightsfor the understanding of the physical mechanisms of the system. For example, thered response functions of vegetation reflect the fact that the short-term productionof vegetation is lower than its long-term balance with precipitation. Therefore, ittakes time for vegetation to fully respond to a precipitation impulse; in otherwords, an impulse of precipitation will generate variations of vegetation in thefollowing months and not simply during the precipitating month. At the same time,because preceding vegetation anomalies also consume water, this response processis also regulated by the persistence rate of vegetation and soil moisture. Dependingon the values of these parameters, the adjustments of the system can either beoscillatory or nonoscillatory, although the parameter space tends to be dominatedby oscillatory behavior.


The model also helps to clarify the role of vegetation in regulating soil moistureand precipitation. Because vegetation represents the principal pathway in whichwater is depleted from soil, from the perspective of soil water balance, vegetation-induced evapotranspiration acts as a negative feedback to help maintain the sta-bility of the system. For the open-loop model, such equilibrium of the system isreached only when the incoming water (precipitation) is exactly balanced by theoutgoing water (evapotranspiration), such that soil moisture becomes invariant andthe long-term relationship between vegetation and precipitation is established.Deviations arising from either precipitation or vegetation anomalies will be re-strained by this negative feedback process to gradually return to their equilibriumstatus. For the closed-loop model, although vegetation prompts the water exchangebetween land and atmosphere at long time scales, it also enhances the diffusion ofwater to the atmosphere and implicitly removes it from the region. For this reason,without increases in incoming water (via precipitation), increases of vegetation(i.e., external disturbances to vegetation) will have negative overall impacts on soilmoisture and precipitation. This result explains why positive vegetation anomaliesat the beginning of the growing season, probably prompted by warmer tempera-tures, may precede drier conditions later in the summer (W1).

Finally, it is important to note that we do not argue that the model developed inthis paper (as well as the physical mechanisms it represents) is the only way thatvegetation influences precipitation variability. Indeed, because the model is mainlydeveloped to provide a physically meaningful explanation for the observed in-traseasonal covariability between vegetation and climate anomalies (W1), it hassimplified many important processes (e.g., through the albedo feedback; Brovkinet al. 1998) that can significantly influence climate variability at long time scales.The statistical results and the stochastic model of this study are also restrained bythe methodologies and, in particular, by the datasets used in the analyses. There-fore, it is recognized that findings of this study need to be further investigated bydedicated observational and process-based approaches in the future.

Acknowledgments. This work was supported in part by the NASA Earth ScienceEnterprise. The views expressed herein are those of the authors and do not necessarilyreflect the views of NASA. The authors are thankful to Dr. Guiling Wang and anotheranonymous reviewer for their constructive comments and suggestions. CMAP precipita-tion data were provided by the NOAA–CIRES ESRL/PSD Climate Diagnostics branch,Boulder, Colorado, from their Web site (http://www.cdc.noaa.gov/). CPC U.S. UNIFIEDPrecipitation data were provided by the NOAA–CIRES ESRL/PSD Climate Diagnosticsbranch, Boulder, Colorado, from their Web site (http://www.cdc.noaa.gov/). We alsowould like to thank Dr. C. J. Tucker for providing the GIMMS NDVI data.

Appendix

Model Derivation1.1. Objective

The model in the paper was directly developed in the form of linear differenceequations to describe relationships among anomalies of vegetation, soil moisture,


and precipitation. However, interactions among these fields (their full values) areknown to be nonlinear and are typically studied using physically based models inthe literature. As such, a question naturally arises: Is there a link between thestochastic model of this study and those physical models? This appendix is in-tended to address this question.

1.2. The general model

In the first step, we want to show that generally, climate–vegetation interactionsin a semiarid environment may be described by the following form:

dV�dt = −a�V, S�V + b�V, S�S, (A1)

dS�dt = P − c�V, S�S. (A2)

P = �S + fp , (A3)

where V, S, and P represent vegetation (e.g., NDVI), soil moisture, and precipi-tation, respectively; fp stands for the external forcing of precipitation; and a, b, c,and � represent the parameters of the system. Note that a, b, and c are treated asnonnegative functions of vegetation and soil moisture. To keep the model simple,� is still assumed to be a constant coefficient (see below for discussions).

Equation (A1) describes vegetation dynamics. It is based on the notion thatvegetation growth is determined by the balance between the loss of biomass(through respiration or metabolism) and the production of vegetation (throughphotosynthesis). Because the environment is arid, it is assumed that vegetationproduction is mainly regulated by soil moisture (Churkina and Running 1998).Therefore, the parameters a and b can be interpreted as the rate of vegetation lossand of production, respectively. This form of the biomass equation is generallyutilized in dynamic vegetation models (e.g., Foley et al. 1996; Dickinson et al.1998). To illustrate with an example, we rewrite the vegetation model in Zeng etal. (1999) as follows:

dV�dt = −V�� + ��1 − e−kL�V��S�, (A4)

where � is understood as the time constant of vegetation, � is the carbon assimi-lation coefficient, k is a photosynthesis coefficient, L(V) is a plant leaf area indexfunction, and �(S) is a function of soil moisture (Zeng et al. 1999). By comparingEquation (A1) and Equation (A4), it is clear that the former represents a generalform of the latter.

Equation (A2) describes the balance of soil moisture. This equation is basedupon the assumption that in the arid environment, soil moisture is mainly balancedby precipitation (P) and evapotranspiration (ET); that is,

dS�dt = P − ET. (A5)

Equation (A5) is generally known as a simple bucket model. Previous studies (e.g.,Mintz and Serafini 1984; Serafini and Sud 1987; Yamaguchi and Shinoda 2002)suggest that evapotranspiration may be modeled as


ET =PET

fS, (A6)

where f is some function of the soil properties, and PET represents the potentialevapotranspiration.

Potential evapotranspiration (PET) over vegetated surface can be modeled usingthe Penman–Monteith equation (Penman 1953; Monteith 1965); however, thisrequires much more information than is available from datasets used here. Instead,we assume that the ideal PET can be optimally approximated by some function ofvegetation and soil moisture [denoted as c(V,S)]; that is,

PET

f� c�V, S�. (A7)

The parameter function c can be understood as the turnover rate of soil moisture.From Equation (A5) to Equation (A7), we get the general equation of soil waterbalance, Equation (A2).

Equation (A3) describes the soil moisture–precipitation feedbacks in the model.It is based upon the notion that precipitation over large regions is supplied by theinflux of water vapor from outside the given region [i.e., fp in Equation (A3)], andby the local water flux evapotranspirated from the soil (Rodrigues-Iturbe et al.1991a). Therefore, the parameter � in Equation (A3) can be interpreted as theproportion of soil moisture that contributes to local precipitation. Because thisparameter describes only a statistical relationship, and because ultimately we wantto represent such a relationship in a linear form, we assume � is a constantcoefficient to keep the equation simple. (In other cases we can use the techniquesdescribed below to linearize this equation.)

1.3. Linearization

To linearize the general model proposed above, we rewrite the system variablesas sums of their long-term means (i.e., climatologies) and deviations from themeans (i.e., anomalies), that is,

�V = V + V�

S = S + S�

P = P + P�

fp = fp + f�p,

(A8)

where V, S, P, and fp represent the climatologies of the system variables, and V�,S�, P�, and f�p represent the corresponding anomalies. Generally, it is assumed thatthere is an equilibrium among the climatologies of the system, such that

�dV�dt = −a�V, S�V + b�V, S�S = 0

dS�dt = P − c�V, S�S = 0

P = �S + fp.

(A9)


This assumption can be made because, simply speaking, once the external climateforcing (e.g., fp) takes its climatological value, the rest of the system variableswould be expected to reach their climatological values as well.

Based on Equation (A8), we also rewrite the general model Equation (A1)–Equation (A3) as follows:

�d�V + V��dt = −a�V + V�, S + S��V + V�� + b�V + V�, S + S��S + S��

d�S + S��dt = �P + P�� − c�V + V�, S + S��S + S��

�P + P�� = ��S + S�� + �fp + f�p�.

(A10)The basic method to linearize Equation (A10) is to 1) expand the parameters a,

b, and c as linear functions around their climatological values (V, S); 2) neglect allthe second-order anomaly terms (V�, S�, and P�); and 3) eliminate the long-termequilibrium relationships [Equation (A9)] from the equations. The first step can bedone with

�a�V + V�, S + S�� a�V, S� +

�a

�V�V,SV� +

�a

�S�V,SS�

b�V + V�, S + S�� b�V, S� +�b

�V�V,SV� +

�b

�S�V,SS�

c�V + V�, S + S�� c�V, S� +�c

�V�V,SV� +

�c

�S�V,SS�.

(A11)

Following the steps described above, these equations give

dV��dt = − aV� + bS�, (A12)dS��dt = P� − cSS� − cVV�, (A13)

P� = �S� + f�p, (A14)where a, b, cS, and cV are constant coefficients (i.e., the Jocobians; Glendinning1994), and are given by

�a = a�V, S� +

�a

�V�V,SV −

�b

�V�V,SS

b = b�V, S� +�b

�S�V,SS −

�a

�S�V,SV

cS = c�V, S� +�c

�S�V,SS

cV =�c

�V�V,SS

. (A15)

Therefore, Equations (A12), (A13), and (A14) represent the linear form of thegeneral model, which describes interactions among anomalies of the system vari-ables.


1.4. Difference equations

To facilitate the discussion, the model derived in the paper (in difference equa-tions) is rewritten here as

V�t = �V�t−1 + �S�t−1, (A16)

S�t = �S�t−1 + P�t −1

�V�t, (A17)

P�t = �S�t−1 + �t . (A18)

The correspondence between Equation (A16)–Equation (A18) and Equation(A12)–Equation (A14) is apparent. However, there are also some subtle differ-ences. For instance, the coefficients of the difference equations are generallydifferent from those of the original differential equations because coefficients inthe discrete time domain implicitly include information about the data-samplingperiods [i.e., the “dt” term in Equations (A12)–(A14)]. Also, Equation (A17)introduces some instantaneous effects of precipitation and vegetation on soil mois-ture, which are not exactly described by Equation (A13). In addition, the time lagof the soil moisture term in Equation (A18) is adjusted. Such adjustments are madebecause the observational data are generally smoothed over a fixed time period,which may affect the time sequences of the relationships among them. For ex-ample, the instantaneous relationships commonly seen in statistical models arerarely described by differential equations. Generally, these adjustments allow themodel to better describe the statistical relationships among the observational dataand at the same time to preserve the physical meaning of the model and itsparameters.

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Documents

Feedbacks of Vegetation on Summertime Climate …cliveg.bu.edu/download/manuscripts/wlwang03.pdftemperature, particularly when time lags are longer than 2 months (W1). That is, vegetation